University of Groningen
A sub-Riemannian Santaló formula with applications to isoperimetric inequalities and first
Dirichlet eigenvalue of hypoelliptic operators
Prandi, Dario; Rizzi, Luca; Seri, Marcello
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Journal of differential geometry DOI:
10.4310/jdg/1549422105
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Publication date: 2019
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Prandi, D., Rizzi, L., & Seri, M. (2019). A sub-Riemannian Santaló formula with applications to isoperimetric inequalities and first Dirichlet eigenvalue of hypoelliptic operators. Journal of differential geometry, 111(2), 339-279. https://doi.org/10.4310/jdg/1549422105
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TO ISOPERIMETRIC INEQUALITIES AND FIRST DIRICHLET EIGENVALUE OF HYPOELLIPTIC OPERATORS
DARIO PRANDI1, LUCA RIZZI2, AND MARCELLO SERI3
Abstract. In this paper we prove a sub-Riemannian version of the classical Santal´o formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. Our construction works under quite general assumptions, satisfied by any sub-Riemannian structure associated with a Riemannian foliation with totally geodesic leaves (e.g. CR and QC manifolds with symmetries), any Carnot group, and some non-equiregular structures such as the Martinet one. A key ingredient is a “reduction procedure” that allows to consider only a simple subset of sub-Riemannian geodesics.
As an application, we derive isoperimetric-type and (p-)Hardy-type inequalities for a compact domain M with piecewise C1,1 boundary, and a universal lower bound for the first Dirichlet eigenvalue λ1(M ) of the sub-Laplacian,
λ1(M ) ≥ kπ2
L2 ,
in terms of the rank k of the distribution and the length L of the longest reduced sub-Riemannian geodesic contained in M . All our results are sharp for the sub-sub-Riemannian structures on the hemispheres of the complex and quaternionic Hopf fibrations:
S1,→ S2d+1−→ CPp d, S3,→ S4d+3−→ HPp d, d ≥ 1,
where the sub-Laplacian is the standard hypoelliptic operator of CR and QC geometries, L = π and k = 2d or 4d, respectively.
1. Introduction and results
Let (M, g) be a compact connected Riemannian manifold with boundary ∂M . Santal´o formula [17,39] is a classical result in integral geometry that describes the Liouville mea-sure µ of the unit tangent bundle U M in terms of the geodesic flow φt : U M → U M .
Namely, for any measurable function F : U M → R we have (1) Z UM F µ = Z ∂M " Z Uq+∂M Z `(v) 0 F (φt(v))dt ! g(v, nq)ηq(v) # σ(q),
where σ is the surface form on ∂M induced by the inward pointing normal vector n, ηq is the Riemannian spherical measure on UqM , Uq+∂M is the set of inward pointing unit
vectors at q ∈ ∂M and `(v) is the exit length of the geodesic with initial vector v. Finally, UM ⊆ U M is the visible set, i.e. the set of unit vectors that can be reached via the geodesic flow starting from points on ∂M .
In the Riemannian setting, (1) allows to deduce some very general isoperimetric in-equalities and Dirichlet eigenvalues estimates for the Laplace-Beltrami operator as showed by Croke in the celebrated papers [19,20,21].
The extension of (1) to the sub-Riemannian setting and its consequences are not straightforward for a number of reasons. Firstly, in sub-Riemannian geometry the geodesic flow is replaced by a degenerate Hamiltonian flow on the cotangent bundle. Moreover, the unit cotangent bundle (the set of covectors with unit norm) is not compact, but rather has the topology of an infinite cylinder. Finally, in sub-Riemannian geometry there is not a
2010 Mathematics Subject Classification. 53C17, 53C65, 35P15, 57R30, 35R03, 53C65.
clear agreement on which is the “canonical” volume, generalizing the Riemannian measure. Another aspect to consider is the presence of characteristic points on the boundary.
In this paper we extend (1) to the most general class of sub-Riemannian structures for which Santal´o formula makes sense. As an application we deduce Hardy-like inequalities, sharp universal estimates on the first Dirichlet eigenvalue of the sub-Laplacian and sharp isoperimetric-type inequalities.
To our best knowledge, a sub-Riemannian version of (1) appeared only in [36] for the three-dimensional Heisenberg group, and more recently in [34] for Carnot groups, where the natural global coordinates allow for explicit computations. As far as other sub-Riemannian structures are concerned, Santal´o formula is an unexplored technique with potential ap-plications to different settings, including CR (Cauchy-Riemann) and QC (quaternionic contact) geometry, Riemannian foliations, and Carnot groups.
1.1. Setting and examples. Let (N, D, g) be a sub-Riemannian manifold of dimension n, where D ⊆ T N is a distribution that satisfies the bracket-generating condition and g is a smooth metric on D. Smooth sections X ∈ Γ(D) are called horizontal. We consider a compact n-dimensional submanifold M ⊂ N with boundary ∂M 6= ∅.
If (N, D, g) is Riemannian, we equip it with its Riemannian volume ωR. In the genuinely
sub-Riemannian case we fix any smooth volume form ω on M (or a density if M is not orientable). In any case, the surface measure σ = ιnω on ∂M is given by the
contrac-tion with the horizontal unit normal n to ∂M . For what concerns the regularity of the boundary, we assume only that ∂M is piecewise C1,1. (See Remark 7 for the Lipschitz case.)
A central role is played by sub-Riemannian geodesics, i.e., curves tangent to D that locally minimize the sub-Riemannian distance between endpoints. In this setting, the geodesic flow1 is a natural Hamiltonian flow φt: T∗M → T∗M on the cotangent bundle, induced by the Hamiltonian function H ∈ C∞(T∗M ). The latter is a non-negative, degenerate, quadratic form on the fibers of T∗M that contains all the information on the sub-Riemannian structure. Length-parametrized geodesics are characterized by an initial covector λ in the unit cotangent bundle U∗M = {λ ∈ T∗M | 2H(λ) = 1}.
A key ingredient for most of our results is the following reduction procedure. Fix a transverse sub-bundle V ⊂ T M such that T M = D ⊕ V. We define the reduced cotangent bundle T∗Mras the set of covectors annihilating V. On T∗Mrwe define a reduced Liouville volume Θr, which depends on the choice of the volume ω on M . These must satisfy the following stability hypotheses:
(H1) The bundle T∗Mr is invariant under the Hamiltonian flow φt;
(H2) The reduced Liouville volume is invariant, i.e. LH~Θr = 0.
This allows to replace the non-compact U∗M with a compact slice U∗Mr := U∗M ∩ T∗Mr, equipped with an invariant measure (see Section4.3). These hypotheses are verified for:
• any Riemannian structure, equipped with the Riemannian volume;
• any sub-Riemannian structure associated with a Riemannian foliation with totally geodesic leaves, equipped with the Riemannian volume. These includes contact, CR, QC structures with transverse symmetries, and also some non-equiregular structures as the Martinet one on R3. See Section5.2;
• any left-invariant sub-Riemannian structure on a Carnot group2, equipped with the Haar volume, see Section5.1.
An interesting example, coming from CR geometry, is the complex Hopf fibration (CHF) S1,→ S2d+1 p−→ CPd, d ≥ 1,
1Abnormal geodesics are allowed, but strictly abnormal ones, not given by the Hamiltonian flow on the
cotangent bundle, do not play any role in our construction.
∂M π(λ) λ γ(`(λ)) M ∂M π(λ2) λ2∈ U/ M π(λ1) λ1∈ UM M
Figure 1. Exit length (left) and visible set (right). Covectors are repre-sented as hyperplanes, the arrow shows the direction of propagation of the associated geodesic for positive time.
where D := (ker p∗)⊥ is the orthogonal complement of the kernel of the differential of the Hopf map w.r.t. the round metric on S2d+1, and the sub-Riemannian metric g is the restriction to D of the round one. Another interesting structure, coming from QC geometry and with corank 3, is the quaternionic Hopf fibration (QHF)
S3 ,→ S4d+3 p−→ HPd, d ≥ 1,
where HPdis the quaternionic projective space of real dimension 4d and the sub-Riemann-ian structure on S4d+3 is defined similarly to its complex version.
1.2. Sub-Riemannian Santal´o formulas. Consider a sub-Riemannian geodesic γ(t)
with initial covector λ ∈ U∗M . The exit length `(λ) ∈ [0, +∞) is the length after which γ leaves M by crossing ∂M . Similarly, ˜`(λ) is the minimum between `(λ) and the cut length c(λ). That is, after length ˜`(λ) the geodesic either loses optimality or leaves M .
The visible unit cotangent bundle UM ⊂ U∗M is the set of unit covectors λ such that `(−λ) < +∞. (See Fig.1.) Analogously, the optimally visible unit cotangent bundle ˜UM is the set of unit covectors such that ˜`(−λ) < +∞.
For any non-characteristic point q ∈ ∂M , we have a well defined inner pointing unit horizontal vector nq ∈ Dq, and U+
q ∂M ⊂ Uq∗M is the set of initial covectors of geodesics
that, for positive time, are directed toward the interior of M .
As anticipated, we do not consider all the length-parametrized geodesics, i.e. all initial covectors λ ∈ Uq∗M ' Sk−1× Rn−k, but a reduced subset Uq∗Mr ' Sk−1. In the following
the suffix r always denotes the intersection with the reduced unit cotangent bundle U∗Mr. We stress the critical fact that U∗Mr is compact, while U∗M never is, except in the Riemannian setting where the reduction procedure is trivial. With these basic definitions at hand, we are ready to state the sub-Riemannian Santal´o formulas.
Theorem 1 (Reduced Santal´o formulas). The visible set UMr and the optimally visible set ˜UMr are measurable. For any measurable function F : U∗Mr → R we have
Z UMrF µ r=Z ∂M " Z Uq+∂Mr Z `(λ) 0 F (φt(λ))dt ! hλ, nqiηqr(λ) # σ(q), (2) Z ˜ UMrF µ r=Z ∂M " Z Uq+∂Mr Z ˜`(λ) 0 F (φt(λ))dt ! hλ, nqiηqr(λ) # σ(q). (3)
In (2)-(3), µr is a reduced invariant Liouville measure on U∗Mr, ηqr is an appropriate smooth measure on the fibers Uq∗Mr and hλ, ·i denotes the action of covectors on vectors. Indeed both include the Riemannian case, where the reduction procedure is trivial and U∗M ' U M since the Hamiltonian is not degenerate.
Remark 1. Hypotheses (H1) and (H2) are essential for the reduction procedure. An unreduced version of Theorem1holds for any volume ω and with no other assumptions but the Lipschitz regularity of ∂M (see Theorem16and Remark7). However, the consequences we present do not hold a priori, as their proofs rely on the summability of certain functions on U∗Mr, generally false on the non-compact U∗M .
1.3. Hardy-type inequalities. For any f ∈ C∞(M ), let ∇Hf ∈ Γ(D) be the horizontal gradient: the horizontal direction of steepest increase of f . It is defined via the identity
(4) g(∇Hf, X) = df (X), ∀X ∈ Γ(D).
Consider all length-parametrized sub-Riemannian geodesic passing through a point q ∈ M , with covector λ ∈ Uq∗M . Set L(λ) := `(λ) + `(−λ); this is the length of the maximal geodesic that passes through q with covector λ.
Proposition 2 (Hardy-like inequalities). For any f ∈ C0∞(M ) it holds
Z M |∇Hf |2ω ≥ kπ 2 |Sk−1| Z M f2 R2ω, (5) Z M |∇Hf |2ω ≥ k 4|Sk−1| Z M f2 r2ω, (6)
where k = rank D and r, R : M → R are: 1 R2(q) := Z U∗ qMr 1 L2η r q, 1 r2(q) := Z U∗ qMr 1 `2η r q, ∀q ∈ M.
We observe that r is the harmonic mean distance from the boundary defined in [23]. One can also consider the following generalization of Proposition2 for Lp(M, ω) norms.
Proposition 3 (p-Hardy-like inequality). Let p > 1 and f ∈ C0∞(M ). Then
Z M |∇Hf |pω ≥ πppCp,k Z M |f |p Rp ω, (7) Z M |∇Hf |pω ≥ p − 1 p p Cp,k Z M |f |p rp ω, (8)
where k = rank D, the constants πp and Cp,k are πp = 2π(p − 1)1/p p sin(π/p) , Cp,k= k |Sk−1| √ π Γ(k+p2 ) 2Γ(1+p2 )Γ(k2 + 1), and rp, Rp : M → R are 1 Rp(q) := Z U∗ qMr 1 Lpη r q, 1 rp(q) := Z U∗ qMr 1 `pη r q, ∀q ∈ M.
1.4. Lower bound for the first Dirichlet eigenvalue. For any given smooth volume ω, a fundamental operator in sub-Riemannian geometry is the sub-Laplacian ∆ω, playing
the role of the Laplace-Beltrami operator in Riemannian geometry. Under the bracket-generating condition, this is an hypoelliptic operator on L2(M, ω). Its principal symbol is (twice) the Hamiltonian, thus the Dirichlet spectrum of −∆ω on the compact manifold M is positive and discrete. We denote it
As a consequence of Proposition2and the min-max principle, we obtain a universal lower bound for the first Dirichlet eigenvalue λ1(M ) on the given domain. Here by universal we mean an estimate not requiring any assumption on curvature or capacity.
Proposition 4 (Universal spectral lower bound). Let L = supλ∈U∗MrL(λ) be the length of the longest reduced geodesic contained in M . Then, letting k = rank D,
(9) λ1(M ) ≥
kπ2 L2 , where we set the r.h.s. to 0 if L = +∞.
Remark 2. In (9), L cannot be replaced by the sub-Riemannian diameter, as M might contain very long (non-minimizing) geodesics, for example closed ones, and L = +∞. See AppendixBfor more details.
In the Riemannian case, as noted by Croke, we attain equality in (9) when M is the hemisphere of the Riemannian round sphere. We prove the following extension to the sub-Riemannian setting.
Proposition 5 (Sharpness of the eigenvalue lower bound). In Proposition 4, in the fol-lowing cases we have equality, for all d ≥ 1:
(i) the hemispheres Sd+ of the Riemannian round sphere Sd;
(ii) the hemispheres S2d+1+ of the sub-Riemannian complex Hopf fibration S2d+1; (iii) the hemispheres S4d+3+ of the sub-Riemannian quaternionic Hopf fibration S4d+3; all equipped with the Riemannian volume of the corresponding round sphere. In all these cases, L = π and λ1(M ) = d, 2d or 4d, respectively. Moreover, the associated eigenfunc-tion is Ψ = cos(δ), where δ is the Riemannian distance from the north pole.
Remark 3. The Riemannian volume of the sub-Riemannian Hopf fibrations coincides, up to a constant factor, with their Popp volume [6,35], an intrinsic smooth measure in sub-Riemannian geometry. This is proved for 3-Sasakian structures (including the QHF) in [38, Prop. 34] and can be proved exactly in the same way for Sasakian structures (including the CHF) using the explicit formula for Popp volume of [6]. For the case (i) ∆ω is the Laplace-Beltrami operator. For the cases (ii) and (iii) ∆ω is the standard sub-Laplacian
of CR and QC geometry, respectively.
In principle, L can be computed when the reduced geodesic flow is explicit. This is the case for Carnot groups, where reduced geodesics passing through the origin are simply straight lines (they fill a k-plane for rank k Carnot groups). It turns out that, in this case, L = diamH(M ) (the horizontal diameter, that is the diameter of the set M measured through left-translations of the aforementioned straight lines). Thus (9) gives an easily computable lower bound for the first Dirichlet eigenvalue in terms of purely metric quantities.
Corollary 6. Let M be a compact n-dimensional submanifold with piecewise C1,1 bound-ary of a Carnot group of rank k, with the Haar volume. Then,
(10) λ1(M ) ≥ kπ
2 diamH(M )2, where diamH(M ) denotes the horizontal diameter of M .
In particular, if M is the metric ball of radius R, we obtain λ1(M ) ≥ kπ2/(2R)2. Clearly (10) is not sharp, as one can check easily in the Euclidean case.
∂M
q
ϑq
M
Figure 2. Visibility angle on a 2D Riemannian manifold. Only the geodesics with tangent vector in the dashed slice go to ∂M .
1.5. Isoperimetric-type inequalities. In this section we relate the sub-Riemannian area and perimeter of M with some of its geometric properties. Since M is compact, the sub-Riemannian diameter diam(M ) can be characterized as the length of the longest optimal geodesic contained in M . Analogously, the reduced sub-Riemannian diameter diamr(M ) is the length of the longest reduced optimal geodesic contained in M . Indeed diamr(M ) ≤ diam(M ).
Consider all reduced geodesics passing through q ∈ M with covector λ. Some of them originate from the boundary ∂M , that is `(−λ) < +∞; others do not, i.e. `(−λ) = +∞. The relative ratio of the lengths of these two types of geodesics (w.r.t. an appropriate measure on Uq∗Mr) is called the visibility angle ϑq ∈ [0, 1] at q (see Definition6). Roughly speaking, if ϑq = 1 then any geodesic passing through q will hit the boundary and, on the opposite, if it is equal to 0 then q is not visible from the boundary (see Fig.2). Similarly, we define the optimal visibility angle ˜ϑq by replacing `(−λ) with ˜`(−λ). Finally, the least visibility angle is ϑ:= infq∈Mϑq , and similarly for the least optimal visibility angle
˜
ϑ := infq∈Mϑ˜q.
Proposition 7 (Isoperimetric-type inequalities). Let ` := sup{`(λ) | λ ∈ Uq∗Mr, q ∈ ∂M } be the length of the longest reduced geodesic contained in M starting from the boundary ∂M . Then (11) σ(∂M ) ω(M ) ≥ C ϑ ` and σ(∂M ) ω(M ) ≥ C ˜ ϑ diamr(M ), where C = 2π|Sk−1|/|Sk| and we set the r.h.s. to 0 if ` = +∞.
The equality in (11) holds for the hemisphere of the Riemannian round sphere, as pointed out in [19]. We have the following generalization to the sub-Riemannian setting.
Proposition 8 (Sharpness of isoperimetric inequalities). In Proposition7, in the following cases we have equality, for all d ≥ 1:
(i) the hemispheres Sd+ of the Riemannian round sphere Sd;
(ii) the hemispheres S2d+1+ of the sub-Riemannian complex Hopf fibration S2d+1; (iii) the hemispheres S4d+3+ of the sub-Riemannian quaternionic Hopf fibration S4d+3; where ω is the Riemannian volume of the corresponding round sphere. In all these cases ϑ = ˜ϑ = 1 and ` = diamr(M ) = π.
We can apply Proposition 7to Carnot groups equipped with the Haar measure. In this case ϑ = ˜ϑ = 1 and ` = diamr(M ) = diamH(M ). Moreover, ω is the Lebesgue volume of Rn and σ is the associated perimeter measure of geometric measure theory [14].
Corollary 9. Let M be a compact n-dimensional submanifold with piecewise C1,1 bound-ary of a Carnot group of rank k, with the Haar volume. Then,
σ(∂M ) ω(M ) ≥ 2π|Sk−1| |Sk| diam H(M ) ,
where diamH(M ) is the horizontal diameter of the Carnot group.
This inequality is not sharp even in the Euclidean case, but it is very easy to compute the horizontal diameter for explicit domains. For example, if M is the sub-Riemannian metric ball of radius R, then diamH(M ) = 2R.
1.6. Remark on change of volume. Fix a sub-Riemannian structure (N, D, g), a com-pact set M with piecewise C1,1 boundary and a complement V such that (H1) holds. Now assume that, for some choice of volume form ω, also (H2) is satisfied, so that we can carry on with the reduction procedure and all our results hold. One can derive the analogous of Propositions2,3,4,7for any other volume ω0 = eϕω, with ϕ ∈ C∞(M ). In all these results, it is sufficient to multiply the r.h.s. of the inequalities by the volumet-ric constant 0 < α ≤ 1 defined as α := max emin eϕϕ, and indeed replace ω with ω0 = eϕω in
Propositions 2 and 3, σ with σ0 = eϕσ in Proposition 7, and the sub-Laplacian ∆ω with ∆ω0 = ∆ω+ hdϕ, ∇H·i in Proposition4. Analogously, one can deal with the Corollaries6 and9 about Carnot groups.
This remark allows, for example, to obtain results for (sub-)Riemannian weighted mea-sures. This is particularly interesting in the genuinely sub-Riemannian setting since, in some cases, the volume satisfying (H2) might not coincide with the intrinsic Popp one. 1.7. Remark on rigidity. The sharpness results of Propositions5 and 8 hold for hemi-spheres of (sub-)Riemannian structures associated with Riemannian submersions of the sphere with totally geodesic fibers, which have been completely classified in [26]. The only case which is not covered in these propositions is the so-called octonionic Hopf fibration (OHF) S7 ,→ S15→ OP1, which to our best knowledge has not yet been studied from the sub-Riemannian point of view, and for which explicit expressions for the sub-Laplacian do not appear in the literature. It is however likely that the sharpness results of Propositions5 and8 hold also for the hemisphere S15
+ of the sub-Riemannian OHF.
Finally, concerning the universal lower bound of Proposition4, Croke proved the follow-ing rigidity result in the Riemannian case (see [19, Thm. 16]). As we already remarked, the lower bound (9) is non-trivial if and only if all geodesics starting from points of M hit the boundary at some finite time (i.e. ϑ = 1). If, furthermore, every such geodesic minimizes distance up to the point of intersection with the boundary (i.e. ˜ϑ = 1), then we have equality in (9) if and only if M is an hemisphere of the round sphere. See also [21] for a more general rigidity result. The following question is thus natural.
Open question. Are the hemispheres of the CHF, QHF, and possibly OHF, the only
domains on compact sub-Riemannian manifolds tamed by a foliation with totally geodesic leaves (see Section5.2) where the lower bound of Proposition 4is attained?
1.8. Afterwords and further developments. Despite its broad range of applications in Riemannian geometry and its Finsler generalizations [42], only a few works used Santal´o formula in the hypoelliptic setting, all of them in the specific case of Carnot groups [34,36] or 3D Sasakian structures [15]. It is interesting to notice that, in [36], Pansu was able to use Santal´o formula in pairs with minimal surfaces to eliminate the diameter term in
Corollary9and obtain his celebrated isoperimetric inequality. In our general setting, this is something worth investigating.
The study of spectral properties of hypoelliptic operators is an active area of research. Many results are available for the complete spectrum of the sub-Laplacian on closed man-ifolds (with no boundary conditions). We recall [8, 10, 11] for the case of SU(2), CHF and QHF. Furthermore, in [18], one can find the spectrum of the “flat Heisenberg case” (a compact quotient of the Heisenberg group) together with quantum ergodicity results for 3D contact sub-Riemannian structures. Lower bounds for the first (non-zero) eigenvalue of the sub-Laplacian on closed foliation, under curvature-like assumptions, appeared in [9] (see also [7] for a more general statement).
Concerning the Dirichlet spectrum on Riemannian manifolds with boundary, a classical reference is [16]. In the sub-Riemannian setting, we are aware of results for the sum of Dirichlet eigenvalues [40] by Strichartz and related spectral inequalities [28] by Hannson and Laptev, both for the case of the Heisenberg group. To our best knowledge, Propo-sition 4 is the first sharp universal lower bound for the first Dirichlet eigenvalue in the sub-Riemannian setting and in particular for non-Carnot structures.
The study of Hardy’s inequalities, already in the Euclidean setting, ranges across the last century and continues to the present day (see [5,12,25] and references therein). The sub-Riemannian case is more recent, for an account of the known result we mention the works for Carnot groups of Capogna, Danielli and Garofalo (see e.g. [13,22]).
Poincar´e inequalities are strictly connected to Hardy’s ones. On this subject the liter-ature is again huge, we already mentioned the works of Croke and Derdzinski concerning the Riemannian case [19, 20, 21]. Finally, see [30] for results on CR and QC manifolds under Ricci curvature assumptions in the spirit of the Lichnerowicz-Obata theorem.
In this paper we focused mostly on foliations, where our results are sharp. For Carnot groups, Corollaries 6 and 9 appeared in [34] and are not sharp. Let us consider for simplicity the 3D Heisenberg group, with coordinates (x, y, z) ∈ R3. A relevant class of domains for the Dirichlet eigenvalues problem are the “Heisenberg cubes” [0, ε] × [0, ε] × [0, ε2], obtained by non-homogeneous dilation of the unit cube [0, 1]3. These represent a fundamental domain for the quotient H3/εΓ of the 3D Heisenberg group H3by the (dilation of the) integer Heisenberg subgroup Γ (a lattice). This is the basic example of nilmanifold, (we thank R. Montgomery for pointing out this example). For these fundamental domains, the first Dirichlet eigenvalue is unknown. However, we mention that for any Carnot group the reduction technique developed here can be further improved leading to a λ1 estimate for cubes, via the technique sketched in AppendixB.)
1.9. Structure of the paper. In Section 2 we recall some basic definitions about sub-Riemannian geometry and sub-Laplacians. In Section 3 we introduce some preliminary constructions concerning integration on vector bundles that we need for the reduction procedure. In Section4we prove the main result of the paper, namely the reduced Santal´o formula. Section 5 is devoted to examples, and contains the general class of structures where our construction can be carried out. Finally, in Section 6 we apply the reduced Santal´o formula to prove Poincar´e, Hardy, and isoperimetric-type inequalities.
2. Sub-Riemannian geometry
We give here only the essential ingredients for our analysis; for more details see [2,35, 37]. A sub-Riemannian manifold is a triple (M, D, g), where M is a smooth, connected manifold of dimension n ≥ 3, D is a vector distribution of constant rank k ≤ n and g is a smooth metric on D. We assume that the distribution is bracket-generating, that is
span{[Xi1, [Xi2, [. . . , [Xim−1, Xim]]]] | m ≥ 1}q= TqM, ∀q ∈ M,
A horizontal curve γ : [0, T ] → R is a Lipschitz continuous path such that ˙γ(t) ∈ Dγ(t)
for almost any t. Horizontal curves have a well defined length `(γ) =
Z T
0
q
g( ˙γ(t), ˙γ(t))dt. Furthermore, the sub-Riemannian distance is defined by:
d(x, y) = inf{`(γ) | γ(0) = x, γ(T ) = y, γ horizontal}.
By the Chow-Rashevskii theorem, under the bracket-generating condition, d is finite and continuous. Sub-Riemannian geometry includes the Riemannian one, when D = T M . 2.1. Sub-Riemannian geodesic flow. Sub-Riemannian geodesics are horizontal curves that locally minimize the length between their endpoints. Let π : T∗M → M be the cotangent bundle. The sub-Riemannian Hamiltonian H : T∗M → R is
H(λ) := 1 2 k X i=1 hλ, Xii2, λ ∈ T∗M,
where X1, . . . , Xk∈ Γ(D) is any local orthonormal frame and hλ, ·i denotes the action of
covectors on vectors. Let σ be the canonical symplectic 2-form on T∗M . The Hamiltonian vector field ~H is defined by σ(·, ~H) = dH. Then the Hamilton equations are
(12) ˙λ(t) = ~H(λ(t)).
Solutions of (12) are called extremals, and their projections γ(t) := π(λ(t)) on M are smooth geodesics. The sub-Riemannian geodesic flow φt∈ T∗M → T∗M is the flow of ~H. Thus, any initial covector λ ∈ T∗M is associated with a geodesic γλ(t) = π ◦ φt(λ), and
its speed k ˙γ(t)k = 2H(λ) is constant. The unit cotangent bundle is U∗M = {λ ∈ T∗M | 2H(λ) = 1}.
It is a fiber bundle with fiber Uq∗M = Sk−1× Rn−k. For λ ∈ Uq∗M , the curve γλ(t) is a
length-parametrized geodesic with length `(γ|[t1,t2]) = t2− t1.
Remark 4. There is also another class of minimizing curves, called abnormal, that might not follow the Hamiltonian dynamic of (12). Abnormal geodesics do not exist in Rie-mannian geometry, and they are all trivial curves in some basic but popular classes of sub-Riemannian structures (e.g. fat ones). Our construction takes in account only the normal sub-Riemannian geodesic flow, hence abnormal geodesics are allowed, but ignored. Some hard open problems in sub-Riemannian geometry are related to abnormal geodesics [1,35,31].
2.2. The intrinsic sub-Laplacian. Let (M, D, g) be a compact sub-Riemannian mani-fold with piecewise C1,1 boundary ∂M , and ω ∈ ΛnM be any smooth volume form (or a
density, if M is not orientable). We define the Dirichlet energy functional as E(f ) =
Z
M
2H(df ) ω, f ∈ C0∞(M ).
The Dirichlet energy functional induces the operator −∆ω on L2(M, ω). Its Friedrichs extension is a non-negative self-adjoint operator on L2(M, ω) that we call the Dirichlet sub-Laplacian. Its domain is the space H01(M ), the closure in the H1(M ) norm of the space C0∞(M ) of smooth functions that vanish on ∂M . Since k∇Hf k2 = 2H(df ), for smooth functions we have
∆ωf = divω(∇Hf ), ∀f ∈ C0∞(M ),
where the divergence is computed w.r.t. ω, and ∇H is the horizontal gradient defined by (4). The spectrum of −∆ω is discrete and positive,
In particular, by the min-max principle we have (13) λ1(M ) = inf E(f ) f ∈ C ∞ 0 (M ), Z M |f |2ω = 1 . 3. Preliminary constructions
We discuss some preliminary constructions concerning integration on vector bundles that we need for the reduction procedure. In this section π : E → M is a rank k vector bundle on an n dimensional manifold M . For simplicity we assume M to be oriented and E to be oriented (as a vector bundle). If not, the results below remain true replacing volumes with densities. We use coordinates x on O ⊂ M and (p, x) ∈ Rk× Rn on U = π−1(O) such that the fibers are Eq0 = {(p, x0) | p ∈ Rk}. In a compact notation we write, in coordinates, dp = dp1∧ . . . ∧ dpk and dx = dx1∧ . . . ∧ dxn.
3.1. Vertical volume forms. Consider the fibers Eq ⊂ E as embedded submanifolds
of dimension k. For each λ ∈ Eq, let Λk(TλEq) be the space of alternating multi-linear
functions on TλEq. The space
Λkv(E) := G
λ∈E
Λk(TλEπ(λ))
defines a rank 1 vector bundle Π : Λkv(E) → E, such that Π(η) = λ if η ∈ Λk(TλEπ(λ)).
To see this, choose coordinates (p, x) ∈ Rk× Rnon U = π−1(O) such that the fibers are Eq0 = {(p, x0) | p ∈ R
k}. Thus the vectors ∂
p1, . . . , ∂pk tangent to the fibers Eq are well
defined. The map Ψ : Π−1(U ) → U × R, defined by Ψ(η) = (Π(η), η(∂p1, . . . , ∂pk)) is a
bijection. Suppose that (U0, p0, x0) is another chart, and similarly Ψ0 : Π−1(U0) → U0× R. Then, on Π−1(U0∩ U ) × R we have Ψ0◦ Ψ−1(λ, α) = (λ, det(∂q0/∂q)). Finally, we apply the vector bundle construction Lemma [32, Lemma 5.5].
Definition 1. A smooth, strictly positive section ν ∈ Γ(Λkv(E)) is called a vertical volume form on E. In particular, the restriction νq := ν|Eq of a vertical volume form defines a
measure on each fiber Eq.
Lemma 10 (Disintegration 1). Fix a volume form Ω ∈ Λn+k(E) and a volume form ω ∈ Λn(M ) on the base space. Then there exists a unique vertical volume form ν ∈ Λkv(E) such that, for any measurable set D0⊆ E and measurable f : D → R,
(14) Z D0f Ω = Z π(D0) " Z D0 q fqνq # ω(q), fq := f |Eq, D 0 q:= Eq∩ D0.
If, in coordinates, Ω = Ω(p, x)dp ∧ dx and ω = ω(x)dx, then ν|(p,x) =
Ω(p, x) ω(x) dp.
Proof. The last formula does not depend on the choice of coordinates (p, x) on E. So we can use this as a definition for ν. Moreover, in coordinates,
Ω|(p,x)= Ω(p, x)dp ∧ dx =
Ω(p, x)
ω(x) dp
∧ (ω(x)dx).
Both uniqueness and (14) follow from the definition of integration on manifolds and Fubini
3.2. Vertical surface forms. Let E0 ⊂ E be a corank 1 sub-bundle of π : E → M . That is, a submanifold E0 ⊂ E such that π|E0 : E0 → M is a bundle, and the fibers Eq0 := π−1(q) ∩ E0 ⊂ Eq are diffeomorphic to a smooth hypersurface C ⊂ Rk. As a matter
of fact, we will only consider the cases in which C is a cylinder or a sphere.
Fix a smooth volume form Ω ∈ Λn+k(E). The Euler vector field is the generator of homogenous dilations on the fibers λ 7→ eαλ, for all α ∈ R. In coordinates (p, x) on E we have e =Pni=1pi∂pi. If e is transverse to E
0 we induce a volume form on E0 by µ := ι eΩ. In this setting, a volume form µ ∈ Λn+k−1(E0) is called a surface form. For any vertical volume form ν ∈ Λkv(E), we define a measure on the fibers Eq0 as ηq = ιeν|Eq. With an
abuse of language, we will refer to such measures as vertical surface forms.
Lemma 11 (Disintegration 2). Fix a surface form µ = ιeΩ ∈ Λn+k−1(E0) and a volume form ω ∈ Λn(M ) on the base space. For any measurable set D ⊆ E0 and measurable f : D → R, (15) Z D f µ = Z π(D) " Z Dq fqηq # ω(q), fq:= f |E0 q, Dq := E 0 q∩ D.
Here, ηq= ιeν|Eq and ν is the vertical volume form on E defined in Lemma10.
Proof. Choose coordinates (p, x) on E. As in the proof of Lemma10 µ|(p,x)= ιeΩ|(p,x)= Ω(p, x)ιedp ∧ dx + (−1)kdp ∧ ιedx = Ω(p, x)ιedp ∧ dx = Ω(p, x) ω(x) ιedp ∧ (ω(x)dx).
Thus (15) holds with η|(p,x)= Ω(p,x)ω(x) ιedp. This, together with the local expression of ν in
Lemma10, yields η = ιeν.
Example 1 (The unit cotangent bundle). We apply the above constructions to E = T∗M and E0 = U∗M . In this case Eq0 = Uq∗M are diffeomorphic to cylinders (or spheres, in the Riemannian case). Moreover, we set Ω = Θ, the Liouville volume form, and µ = ιeΘ, the Liouville surface form.3 One can check that Θ = dµ.
Let ν ∈ Λnv(T∗M ) and η = ιeν as in Lemmas 10 and 11. In canonical coordinates, Θ = dp ∧ dx. Then, if ω = ω(x)dx, ν = 1 ω(x)dp and η = 1 ω(x) n X i=1 (−1)i−1pidp1∧ . . . ∧ddpi∧ . . . ∧ dpn.
Choose coordinates x around q0 ∈ M such that ∂x1|q0, . . . , ∂xk|q0 is an orthonormal basis for the sub-Riemannian distribution Dq0. In the associated canonical coordinates we have
Uq∗0M = {(p, x0) ∈ R2n| p21+ . . . + p2k= 1} ' Sk−1× Rn−k.
In this chart, ηq0 is the (n − 1)-volume form of the above cylinder times 1/ω(x0).
Remark 5. This construction gives a canonical way to define a measure on U∗M and its fibers in the general sub-Riemannian case, depending only on the choice of the volume ω on the manifold M . It turns out that this measure is also invariant under the Hamiltonian flow. Notice though that in the sub-Riemannian setting, fibers have infinite volume.
3 Let ϑ ∈ Λ1
(T∗M ) be the tautological form ϑ(λ) := π∗(λ). The Liouville invariant volume Θ ∈ Λ2n(T∗M ) is Θ := (−1)n(n−1)2 dϑ ∧ . . . ∧ dϑ. In canonical coordinates (p, x) on T∗M we have Θ = dp ∧ dx.
3.3. Invariance. Here we focus on the case of interest where E ⊆ T∗M is a rank k vector sub-bundle and E0 ⊂ E is a corank 1 sub-bundle as defined in Section3.2. We stress that E0 is not necessarily a vector sub-bundle, but typically its fibers are cylinders or spheres. Recall that the sub-Riemannian geodesic flow φt : T∗M → T∗M is the Hamiltonian
flow of H : T∗M → R. Moreover, in our picture, M ⊂ N is a compact submanifold with boundary ∂M of a larger manifold N , with dim M = dim N = n.
Definition 2. A sub-bundle E ⊆ T∗M is invariant if φt(λ) ∈ E for all λ ∈ E and t such
that φt(λ) ∈ T∗M is defined. A volume form Ω ∈ Λn+k(E) is invariant if LH~Ω = 0. Our definition includes the case of interest for Santal´o formula, where sub-Riemannian geodesics may cross ∂M 6= ∅. In other words, E is invariant if the only way to escape from E through the Hamiltonian flow is by crossing the boundary π−1(∂M ). Moreover, if Ω is an invariant volume on an invariant sub-bundle E, then φ∗tΩ = Ω.
Lemma 12 (Invariant induced measures). Let E ⊆ T∗M be an invariant vector bundle with an invariant volume Ω. Let E0 ⊂ E be a corank 1 invariant sub-bundle. Let e be a vector field transverse to E0 and µ = ιeΩ the induced surface form on E0. Then µ is invariant if and only if [ ~H, e] is tangent to E0.
In Example1, E = T∗M and E0= U∗M are clearly invariant; in particular ~H is tangent to E0. By Liouville theorem, Ω = Θ is invariant for any Hamiltonian flow Moreover, if the Hamiltonian H is homogeneous of degree d (on fibers), one checks that [ ~H, e] = −(d − 1) ~H and Lemma12yields the invariance of the Liouville surface measure µ = ιeΘ. In particular this holds in Riemannian and sub-Riemannian geometry, with d = 2.
4. Santal´o formula
4.1. Assumptions on the boundary. Let (N, D, g) be a smooth connected sub-Rie-mannian manifold, of dimension n, without boundary. We focus on a compact n-dimen-sional submanifold M with piecewise C1,1 boundary ∂M .
Let q ∈ ∂M such that the tangent space is well defined. We say that q is a characteristic point if Dq ⊆ Tq∂M . If q is non-characteristic, the horizontal normal at q is the unique
inward pointing unit vector nq∈ Dq orthogonal to Tq∂M ∩ Dq. If q ∈ ∂M is characteristic,
we set nq= 0. We call C(∂M ) the set of characteristic points. The size of C(∂M ) has been
studied in [24,4] under various regularity assumptions on ∂M . We give a self-contained proof of the negligibilty of C(∂M ), which we need in the following. The C1,1 regularity assumption cannot be weakened to C1,α, with 0 < α < 1, as shown in [3, Thm. 1.4].
Proposition 13. Let ∂M be piecewise C1,1. Then, the set of characteristic points C(∂M ) has zero measure in ∂M .
Proof. Without loss of generality we assume that N = Rn and that locally ∂M is the graph of a C1,1 function f : Rn−1 → R. Let also u(x, z) = z − f(x), so that locally ∂M = {(x, z) ∈ Rn−1× R | u(x, z) = 0}.
Let ˜A ⊂ Rn−1 be measurable with positive measure, and let A = {(x, f (x)) | x ∈ ˜A} ⊆ ∂M . We claim that if X, Y are smooth vector fields (not necessarily horizontal), tangent to ∂M a.e. on A, then also [X, Y ] is tangent to ∂M a.e. on A. Notice that X is tangent to ∂M a.e. on A if and only if X(u)(x, f (x)) = 0 for a.e. x ∈ ˜A. Consider the Lipschitz function ξ(x) := X(u)(x, f (x)). As a consequence of coarea formula [27], we have
Z ˜ A |∇ξ(x)| dx = Z R Hn−2( ˜A ∩ ξ−1(t)) dt = 0,
where |∇ξ| is the norm of the Euclidean gradient of ξ : Rn−1 → R, and Hn−2 is the
Hausdorff measure. In particular, ∇ξ = 0 a.e. on ˜A. Since Y is tangent to ∂M a.e. on A, the above identity yields that Y (X(u)) = 0 a.e. on A. A similar argument shows that also
Y (X(u)) = 0 a.e. on A. Since [X, Y ](u)(x, f (x)) = X(Y (u))(x, f (x)) − Y (X(u))(x, f (x)) for a.e. x ∈ Rn−1, we have that [X, Y ] is tangent to ∂M a.e. on A, as claimed.
Assume by contradiction that C(∂M ) has positive measure. In particular, applying the above claim to any pair X, Y ∈ Γ(D), and A = C(∂M ), we obtain that [X, Y ] is tangent to ∂M a.e. on C(∂M ). Since [X, Y ] ∈ Γ(T M ), we can apply the claim a finite number of times, obtaining that any iterated Lie bracket of elements of Γ(D) is tangent to ∂M a.e. on C(∂M ). This contradicts the bracket-generating assumption. 4.2. (Sub-)Riemannian Santal´o formula. For any covector λ ∈ Uq∗M , the exit length `(λ) is the first time t ≥ 0 at which the corresponding geodesic γλ(t) = π ◦ φt(λ) leaves
M crossing its boundary, while ˜`(λ) is the smallest between the exit and the cut length along γλ(t). Namely
`(λ) = sup{t ≥ 0 | γλ(t) ∈ M },
˜
`(λ) = sup{t ≤ `(λ) | γλ|[0,t] is minimizing}.
We also introduce the following subsets of the unit cotangent bundle π : U∗M → M : U+∂M = {λ ∈ U∗M |∂M | hλ, ni > 0} ,
UM = {λ ∈ U∗M | `(−λ) < +∞}, ˜
UM = {λ ∈ UM | ˜`(−λ) = `(−λ)}.
Some comments are in order. The set U+∂M consists of the unit covectors λ ∈ π−1(∂M ) such that the associated geodesic enters the set M for arbitrary small t > 0. The visible set UM is the set of covectors that can be reached in finite time starting from π−1(∂M ) and following the geodesic flow. If we restrict to covectors that can be reached optimally in finite time, we obtain the optimally visible set ˜UM (see Fig.1).
Lemma 14. The cut-length c : U∗M → (0, +∞] is upper semicontinuous (and hence measurable). Moreover, if any couple of distinct points in M can be joined by a minimizing non-abnormal geodesic, c is continuous.
Proof. The result follows as in [17, Thm. III.2.1]. We stress that the key part of the proof of the second statement is the fact that, in absence of non-trivial abnormal minimizers, a point is in the cut locus of another if and only if (i) it is conjugate along some minimizing geodesic or (ii) there exist two distinct minimizing geodesics joining them.
Lemma 15. The exit length ` : U+∂M → (0, +∞] is lower semicontinuous (and hence measurable). Moreover, ˜` : U+∂M → (0, +∞] is measurable.
Proof. Let λ0 ∈ U+∂M . Consider a sequence λ
n such that lim infλ→λ0`(λ) = limn`(λn). Then, the trajectories γn(t) = π ◦ φt(λn) for t ∈ [0, `(λn)] converge uniformly as n → +∞
to the trajectory γ0(t) = φt(λ0) for t ∈ [0, δ] where δ = limn`(λn). Moreover, by continuity
of ∂M and the fact that γn(`(λn)) ∈ ∂M , it follows that γ0(δ) ∈ ∂M . This proves that δ ≥ `(λ0), proving the first part of the statement.
To complete the proof, observe that ˜` = min{`, c}, which are measurable by the previous
claim and Lemma14.
Fix a volume form ω on M (or density, if M is not orientable). In any case, ω and σ := ιnω induce positive measures on M and ∂M , respectively. According to Lemmas10
Theorem 16 (Santal´o formulas). The visible set UM and the optimally visible set ˜UM are measurable. Moreover, for any measurable function F : U∗M → R we have
Z UM F µ = Z ∂M " Z Uq+∂M Z `(λ) 0 F (φt(λ))dt ! hλ, nqiηq(λ) # σ(q), (16) Z ˜ UM F µ = Z ∂M " Z Uq+∂M Z `(λ)˜ 0 F (φt(λ))dt ! hλ, nqiηq(λ) # σ(q). (17)
Remark 6. Even if M is compact and hence ˜` < +∞, in general ˜UM ( UM . Fur-thermore, if ` < +∞ (that is, all geodesics reach the boundary of M in finite time), then UM = U∗M . Thus, our statement of Santal`o formula contains [17, Thm. VII.4.1]. Remark 7. If ∂M is only Lipschitz and C(∂M ) has positive measure, the above Santal´o formulas still hold by removing on the left hand side from UM and ˜UM the set {φt(λ) |
π(λ) ∈ C(∂M ) and t ≥ 0}. Nothing changes on the right hand side as σ(C(∂M )) = 0, since σ = ιnω and n vanishes on C(∂M ) by definition.
Proof. Let A ⊂ [0, +∞) × U+∂M be the set of pairs (t, λ) such that 0 < t < `(λ). By Lemma 15 it follows that A is measurable. Let also Z = π−1(∂M ) ⊂ UM which has zero measure in U∗M . Define φ : A → UM \ Z as φ(t, λ) = φt(λ). This is a smooth
diffeomorphism, whose inverse is φ−1(¯λ) = (`(−¯λ), −φ`(−¯λ)(−¯λ)). In particular, UM is measurable. Then, using Lemma17 (see below), and Fubini theorem, we have
(18) Z UM F µ = Z φ(A) F µ = Z A (F ◦ φ) φ∗µ = = Z ∂M " Z Uq+∂M Z `(λ) 0 F (φt(λ))dt ! hλ, nqiηq(λ) # σ(q), which proves (16). Analogously, with ˜A = {(t, λ) | 0 < t < ˜`(λ)} and ˜Z = Z ∪ {φ˜`(λ)(λ) | λ ∈ U+∂M } the map φ : ˜A → ˜UM \ ˜Z is a diffeomorphism with the same inverse. Then, the same computations as (18) replacing A with ˜A and Z with ˜Z yield (17).
Lemma 17. The following local identity of elements of Λ2n−1(R × U+∂M ) holds φ∗µ|(t,λ) = hλ, nqi dt ∧ σ ∧ η, λ ∈ U+∂M,
where, in canonical coordinates (p, x) on T∗M η = ιeν, ν =
1
ω(x)dp, σ = ιnω, ω = ω(x)dx.
Proof. For any (t, λ) ∈ R × U+∂M let {∂t, v1, . . . , v2n−2} be a set of independent vectors
in T (R × U+∂M ) = T R ⊕ T U+∂M . Observe that φ∗µ = dt ∧ (ι ∂tφ ∗µ). Then, ι∂tφ ∗µ(v 1, . . . , v2n−2) = µ|φ(t,λ) d(t,λ)φ ∂t, d(t,λ)φ v1, . . . , d(t,λ)φ v2n−2 . Notice that,
(a) d(t,λ)φ ∂t= (dλφt) ~H, this is in fact just ~H|φt(λ),
(b) d(t,λ)φ vi = (dλφt)vi for any i = 1, . . . , 2n − 2.
Hence it follows that
(19) ι∂tφ ∗µ = ι ~ Hφ ∗ tµ = ιH~µ,
where in the last passage we used the invariance of µ (see the discussion below Lemma12). By Lemma 11 and its proof (in particular see Example 1) locally µ = η ∧ ω. By the properties of the interior product,
The first term on the r.h.s. vanishes: as a 2n − 2 form, its value at a point λ ∈ U+∂M is completely determined by its action on 2n − 2 independent vectors of TλU+∂M . We
can choose coordinates such that ∂M = {xn= 0}. Then a basis of TλU+∂M is given by ∂x1, . . . , ∂xn−1 and a set of n − 1 vectors vi =
Pn j=1v j i∂pj in TλU + π(λ)∂M . Since ιH~η is a
n − 2 form, then ω necessarily acts on at least one vi, and vanishes. Now, notice that
(21) ιH~ω|λ(·) = ω|π(λ)(π∗H, π∗~ ·) = hλ, nπ(λ)i ω|π(λ)(nπ(λ), π∗·) = hλ, nπ(λ)iσ|λ(·).
Putting together (19), (20), and (21) completes the proof of the statement. 4.3. Reduced Santal´o formula. The following reduction procedure replaces the
non-compact set UM in Theorem16 with a compact subset that we now describe.
To carry out this procedure we fix a transverse sub-bundle V ⊂ T M such that T M = D⊕V. We assume that V is the orthogonal complement of D w.r.t. to a Riemannian metric g such that g|D coincides with the sub-Riemannian one and the associated Riemannian volume coincides with ω. In the Riemannian case, where V is trivial, this forces ω = ωR, the
Riemannian volume. In the genuinely sub-Riemannian case there is no loss of generality since this assumption is satisfied for any choice of ω.
Definition 3. The reduced cotangent bundle is the rank k vector bundle π : T∗Mr → M of covectors that annihilate the vertical directions:
T∗Mr:= {λ ∈ T∗M | hλ, vi = 0 for all v ∈ V} . The reduced unit cotangent bundle is U∗Mr := U∗M ∩ T∗Mr.
Observe that U∗Mr is a corank 1 sub-bundle of T∗Mr, whose fibers are spheres Sk−1. If T∗Mr is invariant in the sense of Definition2, we can apply the construction of Section3.3. The Liouville volume Θ on T∗M induces a volume on T∗Mr as follows.
Let X1, . . . , Xkand Z1, . . . , Zn−k be local orthonormal frames for D and V, respectively.
Let ui(λ) := hλ, Xi) and vj(λ) := hλ, Zji smooth functions on T∗M . Thus T∗Mr = {λ ∈ T∗M | v1(λ) = . . . = vn−k(λ) = 0}.
For all q ∈ M where the fields are defined, (u, v) : Tq∗M → Rn are smooth coordinates on the fiber and hence ∂u1, . . . , ∂uk,∂v1, . . . , ∂vn−k are vectors on Tλ(T
∗
qM ) ⊂ Tλ(T∗M ) for all
λ ∈ π−1(q). In particular, the vector fields ∂v1, . . . , ∂vn−k are transverse to T
∗Mr, hence we give the following definition.
Definition 4. The reduced Liouville volume Θr ∈ Λn+k(T∗Mr) is Θrλ:= Θλ(. . . , . . . , . . . | {z } k vectors , ∂v1, . . . , ∂vn−k, . . . , . . . , . . . | {z } n vectors ), ∀λ ∈ T∗Mr.
The above definition of Θr does not depend on the choice of the local orthonormal frame {X1, . . . , Xk, Z1, . . . , Zn−k} and Riemannian metric g|V on the complement, as long
as its Riemannian volume remains the fixed one, ω. In fact, let X0, Z0 be a different frame for a different Riemannian metric g0|V. Then4, X0 = RX and Z0 = SX + T Z for R ∈ SO(k), T ∈ SL(n − k) and S ∈ M(k, n). One can check that ∂v = S∂u0 + T ∂v0 and that Θr = Θ(. . . , ∂v, . . .) = Θ(. . . , T ∂v0, . . .) = (Θr)0, where both frames are defined.
Assumptions for reduction. We assume the following hypotheses:
(H1) The bundle T∗Mr ⊆ T∗M is invariant.
(H2) The reduced Liouville volume is invariant, i.e. LH~Θr = 0.
Remark 8. Assumption (H1) depends only on V, while (H2) depends also on ω (since Θr does). In the Riemannian case, with ω = ωR, both are trivially satisfied.
4For simplicity, assume that D is orientable as a vector bundle and that X
Under these assumptions U∗Mr = U∗M ∩ T∗Mr is an invariant corank 1 sub-bundle of T∗Mr. Moreover, µr = ιeΘr is an invariant surface form on U∗Mr. This follows from Lemma12observing that [ ~H, e] = − ~H is tangent to U∗Mr. As in Section3.1, the volume Θr ∈ Λn+k(T∗Mr) induces a vertical volume νr
q on the fibers Tq∗Mr and a vertical surface
form ηqr = ιeνq on Uq∗Mr. As a consequence of Lemmas 10 and 11 the latter has the
following explicit expression, whose proof is straightforward.
Lemma 18 (Explicit reduced vertical measure). Let q0 ∈ M and fix a set of canonical coordinates (p, x) such that q0 has coordinates x0 and
• {∂x1, . . . , ∂xk}q0 is an orthonormal basis of Dq0, • {∂xk+1, . . . , ∂xn}q0 is an orthonormal basis of Vq0. In these coordinates ω|x0 = dx|q0. Then νqr
0 = volRk and η r q0 = volSk−1. In particular, Z U∗ q0Mr ηrq0 = |Sk−1|, ∀q0 ∈ M, where |Sk−1| denotes the Lebesgue measure of Sk−1and vol
Rk, volSk−1 denote the Euclidean volume forms of Rk and Sk−1.
We now state the reduced Santal´o formulas. The sets U+∂Mr, UMr, and ˜UMr are defined from their unreduced counterparts by taking the intersection with T∗Mr.
Theorem 19 (Reduced Santal´o formulas). The visible set UMr and the optimally visible set ˜UMr are measurable. For any measurable function F : U∗Mr → R we have
Z UMrF µ r=Z ∂M " Z Uq+∂Mr Z `(λ) 0 F (φt(λ))dt ! hλ, nqiηqr(λ) # σ(q), (22) Z ˜ UMrF µ r=Z ∂M " Z Uq+∂Mr Z ˜`(λ) 0 F (φt(λ))dt ! hλ, nqiηqr(λ) # σ(q).
Proof. The proof follows the same steps as the one of Theorem16replacing the invariant sub-bundles, volumes, and surface forms with their reduced counterparts. Remark 9. Let HsR be the sub-Riemannian Hamiltonian and HR be the Riemannian
Hamiltonian of the Riemannian extension. The two Hamiltonians are (locally on T∗M ) HR= 1 2 k X i=1 u2i + n−k X j=1 v2j , HsR = 1 2 k X i=1 u2i. Let φsRt = et ~HsR and φR
t = et ~HR be their Hamiltonian flows. Since T∗Mr = {λ | v1(λ) = . . . = vn−k(λ) = 0}, by assumption (H1) we have
HsR = HR, and φtsR= φRt on T∗Mr.
In particular, the sub-Riemannian geodesics with initial covector λ ∈ U∗Mr are also geodesics of the Riemannian extension and viceversa.
5. Examples
5.1. Carnot groups. A Carnot group (G, ?) of step m is a connected, simply connected Lie group of dimension n, such that its Lie algebra g = TeG admits a nilpotent stratification
of step m, that is
g = g1⊕ . . . ⊕ gm, with
Let D be the left-invariant distribution generated by g1, and consider any left-invariant sub-Riemannian structure on G induced by a scalar product on g1.
We identify G ' Rn with a polynomial product law by choosing a basis for g as follows. Recall that the group exponential map,
expG: g → G,
associates with V ∈ g the element γ(1), where γ : [0, 1] → G is the unique integral line starting from γ(0) = 0 of the left invariant vector field associated with V . Since G is simply connected and g is nilpotent, expG is a smooth diffeomorphism.
Let dj := dim gj. Indeed d1 = k. Let {Xij}, for j = 1, . . . , m and i = 1, . . . , dj be an
adapted basis, that is gj = span{X1j, . . . , X
j
dj}. In exponential coordinates we identify
(x1, . . . , xm) ' expG m X j=1 dj X i=1 xjiXij , xj ∈ Rdj.
The identity e ∈ G is the point (0, . . . , 0) ∈ Rn and, by the Baker-Cambpell-Hausdorff formula the group law ? is a polynomial expression in the coordinates (x1, . . . , xm). Finally,
Xij = ∂ ∂xji 0 ,
so that D|e ' {(x, 0, . . . , 0) | x ∈ Rk} and Dq = Lq∗D|e, where Lq∗ is the differential of the
left-translation Lq(p) := q ? p.
We equip G with the Lebesgue volume of Rn, which is a left-invariant Haar measure. In order to apply the reduction procedure of Section4.3, let V be the left-invariant distri-bution generated by
V|e:= g2⊕ . . . ⊕ gm,
and consider any left-invariant scalar product g|V on V. Thus, up to a renormalization, g = g|D ⊕ g|V is a left-invariant Riemannian extension such that T M = D ⊕ V is an orthogonal direct sum and its Riemannian volume coincides with the Lebesgue one.
Proposition 20. Any Carnot group satisfies assumptions (H1) and (H2).
Proof. Let X1, . . . , Xk ∈ Γ(D) and Z1, . . . , Zn−k ∈ Γ(V) be a global frame of left-invariant
orthonormal vector fields. Let ui(λ) := hλ, Xii and vj(λ) := hλ, Zji be smooth functions
on T∗G. We have the following expressions for the Poisson brackets {ui, vj} = k X i=1 n−k X `=1 d`ijv`, i = 1, . . . , k, j = 1, . . . , n − k,
for some constants d`ij. We stress that the above expression does not depend on the ui’s, as
a consequence of the graded structure. Denoting the derivative along the integral curves of ~H with a dot, we have
˙vj = {H, vj} = k X i=1 ui{ui, vj} = k X i=1 n−k X `=1 uid`ijv`.
Thus, any integral line of ~H starting from λ ∈ T∗Mr = {v1= . . . = vn−k= 0} remains in
T∗Mr and the latter is invariant.
To prove the invariance of Θr, consider, for any fixed left-invariant X ∈ Γ(D), the adjoint map adX : V|e → V|e, given by adX(Z) = [X, Z]|e. This map is well defined (as a
consequence of the graded structure) and nilpotent. In particular Trace(adX) = 0. Thus, we obtain from an explicit computation (see AppendixA)
LH~Θr = − k X i=1 n−k X j=1 uidjij Θr= − k X i=1 uiTrace(adXi) ! Θr = 0.
Proposition 21 (Characterization of reduced geodesics for Carnot groups). The geodesics
γλ(t) with initial covector λ ∈ Tq∗Mr are obtained by left-translation of straight lines, that
is, in exponential coordinates,
γλ(t) = q ? (ut, 0, . . . , 0), u ∈ Rk.
Proof. Let X1, . . . , Xk ∈ Γ(D) and Z1, . . . , Zn−k ∈ Γ(V) be a global frame of left-invariant
orthonormal vector fields. Let ui(λ) := hλ, Xii and vj(λ) := hλ, Zji be smooth functions
on T∗G. Let u ∈ Rk. The extremal λ(t) = φt(λ), with initial covector λ = (q, u, 0) satisfies
v ≡ 0 by Proposition20 and, as a consequence of the graded structure, ˙ ui = {H, ui} = k X j=1 uj{uj, ui} = k X j=1 n−k X `=1 ujc`jiv`= 0,
In particular λ(t) = (q(t), u, 0). Moreover the geodesic γλ(t) = π(λ(t)) satisfies ˙γλ(t) =
k X
i=1
uiXi(γλ(t)).
Since the ui’s are constants, γλ(t) is an integral curve ofPki=1uiXi starting from q. Then
L−1q γλ(t) is an integral curve of Pki=1uiL−1q∗Xi =Pki=1uiXi starting from the identity. By
definition of exponential coordinates
γλ(t) = q ? expG t k X i=1 uiXi ! ' q ? (ut, 0).
Remark 10. In the case of a step 2 Carnot group, the group law is linear when written in exponential coordinates. In fact, for a fixed left-invariant basis X1, . . . , Xk∈ Γ(D) and
Z1, . . . , Zn−k ∈ Γ(V) it holds [Xi, Xj] = n−k X `=1 c`ijZ`, c`ij ∈ R.
By the Baker-Campbell-Hausdorff formula, (x, z) ? (x0, z0) = (x0 + x, z0 + z + f (x, x0)), where f (x, x0)`= 1 2 k X i,j=1 xic`ijx 0 j, ` = 1, . . . , n − k.
As a consequence, the geodesics γλ(t) with initial covector λ ∈ Tq∗Mr span the set q ? D|e.
The latter is not an hyperplane, in general, when q 6= e and the step m > 2.
Example 2 (Heisenberg group). The (2d + 1)-dimensional Heisenberg group H2d+1 is the sub-Riemannian structure on R2d+1 where (D, g) is given by the following set of global orthonormal fields Xi:= ∂xi− 1 2 2d X i=1 Jijxj∂z, J = 0 In −In 0 , i = 1, . . . , 2d,
written in coordinates (x, z) ∈ R2d × R. The distribution is bracket-generating, as [Xi, Xj] = Jij∂z. These fields generate a stratified Lie algebra, nilpotent of step 2, with
There is a unique connected, simply connected Lie group G such that g = g1 ⊕ g2 is its Lie algebra of left-invariant vector fields. The group exponential map expG : g → G is a smooth diffeomorphism and then we identify G = R2d+1 with the polynomial product law
(x, z) ? (x0, z0) =
x + x0, z + z0+1 2x · J x
0.
Notice that X1, . . . , X2d (and ∂z) are left-invariant.
To carry on the reduction, we consider the Riemannian extension g such that ∂z is a unit vector orthogonal to D. The geodesics associated with λ ∈ U∗Mr and starting from q reach the whole Euclidean plane q ? {z = 0} (the left-translation of R2d ⊂ R2d+1). At q = (x, z) this is the plane orthogonal to the vector12J x, 1w.r.t. the Euclidean metric. 5.2. Riemannian foliations with bundle like metric. Roughly speaking, a Riemann-ian foliation has bundle like metric if locally it is a RiemannRiemann-ian submersion w.r.t. the projection along the leaves.
Definition 5. Let M be a smooth and connected n-dimensional Riemannian manifold.
A k-codimensional foliation F on M is said to be Riemannian with bundle like metric if there exists a maximal collection of pairs {(Uα, πα), α ∈ I} of open subsets Uα of M and
submersions πα: Uα→ U0
α ⊂ Rk such that
• {Uα}α∈I is a covering of M
• If Uα ∩ Uβ 6= ∅, there exists a local diffeomorphism Ψαβ : Rk → Rk such that
πα = Ψαβπβ on Uα∩ Uβ
• the maps πα : Uα → U0
α are Riemannian submersions when Uα0 are endowed with
a given Riemannian metric
On each Uα, the preimages πα−1(x0) for fixed x0 ∈ U0
α are codimension k embedded
sub-manifolds, called the plaques of the foliation. These submanifolds form maximal connected injectively immersed submanifolds called the leaves of the foliation. The foliation is totally geodesic if its leaves are totally geodesic submanifolds [41].
To any Riemannian foliation with bundle-like metric we associate the splitting T M = D ⊕ V, where V is the bundle of vectors tangent to the leaves of the foliation and D is its orthogonal complement (we call V the bundle of vertical directions, and its sections vertical vector fields). If D is bracket-generating, then (D, g|D) is indeed a sub-Riemannian structure on M that we refer to as tamed by a foliation and we assume to be equipped with the corresponding Riemannian volume.
We say that a vector field X ∈ Γ(T M ) is basic if, locally on any Uα, it is πα-related
with some vector X0 on Uα0. If X ∈ Γ(T M ) is basic, and V ∈ Γ(V) is vertical, then the Lie bracket [X, V ] is vertical. In this setting we consider a local orthonormal frame Z1, . . . , Zn−k ∈ Γ(V) of vertical vector fields and a local orthonormal frame of basic vector
fields X1, . . . , Xk∈ Γ(D) for the distribution. The structural functions are defined as
[Xi, Xj] = k X `=1 b`ijX`+ n−k X `=1 c`ijZ`, [Xi, Zj] = n−k X `=1 d`ijZ`, [Zi, Zj] = n−k X `=1 e`ijZ`.
The totally geodesic assumption is equivalent to the fact that any basic horizontal vector field X generates a vertical isometry, that is
(23) (LXg)(Z, W ) = 0, ∀Z, W ∈ Γ(V) ⇐⇒ d`ij = −dji`.
Proposition 22. Any sub-Riemannian structure tamed by a foliation with totally geodesic
leaves satisfies assumptions (H1) and (H2).
Proof. Locally, T∗Mr is the zero-locus of the functions vi(λ) = hλ, Zii for some family
